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arXiv 0901.0791gr-qchep-thmath-phmath.MP
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以标量曲率不变量表征的时空

Spacetimes characterized by their scalar curvature invariants

Alan Coley, Sigbjorn Hervik, Nicos Pelavas

摘要:

在本文中,我们确定了一类四维洛伦兹流形,可以完全由黎曼张量及其协变导数构造的标量多项式曲率不变量的特点。我们引入了$\mathcal{I}$ -非退化时空度规的概念,这意味着时空度规是由其曲率不变量局部决定的。通过从Riemann张量及其协变导数中确定一组合适的投影算子,我们能够证明时空度规是$\mathcal{I}$ -非退化时的一些结果(包括代数上的一般和代数上的特殊情况).这使我们能够证明我们的主要定理,时空度规是$\mathcal{I}$ -非简并的或Kundt度规。因此,一个不以曲率不变量为特征的度量必须是退化的孔特形式。然后,我们讨论了什么性质的基本时空可以确定从一组给定的标量多项式不变量的逆问题,并提出了一些部分结果。我们还讨论了\emph{strong}和\emph{weak}非简并的概念。

英文摘要:

In this paper we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an $\mathcal{I}$-non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is $\mathcal{I}$-non-degenerate. This enables us to prove our main theorem that a spacetime metric is either $\mathcal{I}$-non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of \emph{strong} and \emph{weak} non-degeneracy.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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